**Abstract**

In this chapter, interlinkages between stock markets in CEE-4 countries and capital markets in developed countries are analyzed. Changes of variance on stock markets in Poland, the Czech Republic, Slovakia, and Hungary are identified. Differences among countries are analyzed. Capital markets of these countries are compared in terms of market efficiency. Moreover, co-movements of stock markets in Visegrad countries with capital markets in developed countries are studied. Different specifications of multivariate GARCH models are studied. Asymmetric GARCH-BEKK model and Asymmetric Generalized Dynamic Conditional Correlation model are considered.

**Keywords:** stock markets, Visegrad countries, market efficiency, multivariate GARCH, shocks' transmission

### **1. Introduction**

The Visegrad countries are the four Central European Countries (referred to as the CEE-4 henceforth)—Czech Republic, Poland, Slovakia, and Hungary. These countries joined the European Union (EU) in spring 2004, and three of them are still (January 2020) committed to adopting the euro at some point. At the early stages of transformation, their stock markets were relatively poorly integrated with the stock markets of the EU countries (the so-called old EU countries). This is due to the shorter history of the free-market economy in Poland, Hungary, and the Czech Republic. Low level of correlation between rates of return on stock market indexes in CEE-4 countries and rates of return on stock market indexes of industrialized economies resulted in considering assets from post-communist economies in investors' portfolios (e.g., see [1, 2]). The accession of Poland, Hungary, and the Czech Republic to the EU on May 1, 2004 attracted the interest of numerous investors who had not earlier invested in these countries due to political, corporate governance, and liquidity risks (see [3]). In particular, the Slovakian stock market was strongly integrated with capital markets of developed economies due to the country's participation in the Exchange Rate Mechanism 2 (ERM2 henceforth) and euro adoption in 2009 (see [4]). A significant increase in the level of integration resulted in a decrease in the benefits from using the portfolio diversification strategy.

During the US subprime crisis, a significant increase in correlations between stock markets in CEE-3 countries and capital markets in developed economies was observed (see [5]). In particular, correlation between shocks in the Slovakian stock market and shocks in industrialized countries was very large. The outbreak of the euro area sovereign debt crisis resulted in a slight decrease in the level of correlation between stock markets in CEE-4 countries and mature capital markets. However, rates of return in the German stock market still strongly affected the rates of return in the Polish, Hungarian, Czech, and Slovakian stock market. After the introduction of the Outright Monetary Transactions program (OMT henceforth), the sensitivity of the stock market returns in Poland, the Czech Republic, and Hungary to external shocks decreased significantly. This phenomenon was not observed in the case of Slovakia, which is a member of the euro area.

integration resulted in a significant decrease in the benefits from using the portfolio

*Stock Markets of the Visegrad Countries after Their Accession to the European Union*

The impact of the subprime crisis on the performance of stock markets in CEE-4 countries has been broadly discussed in the economic literature. Very significant transmission of shocks to CEECs'stock markets during the subprime crisis was identified in numerous research studies (see, e.g., [17–19]). As Syllignakis and Kouretas [13] suggested, the contagion transmission from the major stock markets to capital markets of the CEECs in the period of financial turmoil as well as during the subprime crisis was due to increased financial liberalization and increased

In turn, studies devoted to the role of the sovereign debt crisis in shaping stock market prices in the Central and Eastern Europe seem to be rare. The analysis of dynamic correlation coefficients conducted by Bein and Tuna [11] indicates that during the euro-area sovereign debt crisis, stock markets in the analyzed region have been highly correlated with the finance-led markets of GIIPS (Greece, Italy, Ireland, Portugal, and Spain) as well as with stock markets of the EU3 (France, Germany, and the United Kingdom). Moreover, significant spillover effect from capital markets of Italy, Ireland, Portugal, and Spain to stock markets of the Central and Eastern European countries has been identified. Differences in the level of reaction of stock markets in CEE-4 countries during the euro area sovereign debt crisis have been noticed as well. Due to the fact that Slovakia was a member of the euro area in 2010–2011, stock market in this country reacted stronger to positive and negative shocks. Moreover, the Polish stock market has shown a significantly higher level of conditional correlation than the Czech Republic and Hungary. Some results in the literature devoted to the sensitivity of capital markets in Visegrad countries to external shocks have pointed out their asymmetric reaction (see, e.g., [18, 20]). This finding has been interpreted as a problem of information asymmetry

As Grabowski [20] and Moagar-Poladian et al. [22] have noticed, in the period of

In order to identify different states of stock markets, the methodology proposed

moments of significant changes in the unconditional variance were identified. In formula (1), the number of observations denoted by *T* and *Dk* <sup>∗</sup> is defined as follows:

ffiffiffiffiffiffiffiffi

*T=*2 p (1)

*<sup>k</sup>* j j *Dk* , (2)

by Inclan and Tiao [7] and Inclan et al. [24] should be used. On the basis of the

*IT* ¼ *Dk* <sup>∗</sup>

*Dk* <sup>∗</sup> <sup>¼</sup> *max*

financial stability (2013–2019) in the case of Poland, the Czech Republic, and Hungary, the sensitivity of the stock market returns to external shocks became meaningfully weaker. In the case of the Slovakian stock market, drop in the level of integration was smaller (see, e.g., [23]). Moreover, after 2012, the evolution of the stock market indices in the CEE-4 countries has followed different paths. As a result of the lower level of market uncertainty, volatility spillovers have weakened. A within-group integration of stock markets of the CEE-4 countries has decreased significantly. Opportunities for portfolio diversification have increased with discrepancies between market returns observed after 2012 (see. [20]).

diversification strategy (see [16]).

*DOI: http://dx.doi.org/10.5772/intechopen.92102*

participation of foreign investors in these markets.

and the presence of agents with superior knowledge [21].

**3. Methodology**

following statistics,

**139**

Nevertheless, we still think that we can contribute to this body of literature. Previous studies devoted to the analysis of the performance of stock markets in Poland, the Czech Republic, Slovakia, and Hungary compared sensitivity of these markets to external and internal shocks in a priori defined subperiods. In most cases, the Winkler's [6] periodization of the global financial crisis was used. In this research study, we identify days of statistically significant breakpoints on the basis of the method proposed by Inclan and Tiao [7]. Stock markets of Poland, the Czech Republic, Slovakia, and Hungary are compared with regard to timing of significant breakpoints. Moreover, efficiency of stock markets in CEE-4 countries is compared across subperiods and across markets. Sensitivity of stock markets to external shocks is compared in the context of the volatility transmission and linkages between rates of return. Differences between resistance to shocks from the United States and Germany (main economy of the European Union) are analyzed as well. The obtained differences between Slovakia and noneuro-area member states should provide recommendations for policy makers of Poland, the Czech Republic, and Hungary in the context of the future accession of these countries to the euro area.

This chapter has the following structure. In Section 2, the literature review is provided. In Section 3, the methodology is presented. In Section 4, findings from the empirical research are presented and discussed. The last section concludes the study.

#### **2. Literature review**

Poor integration of capital markets in Poland, the Czech Republic, Hungary, and Slovakia in the twentieth century resulted in low level of interest in sensitivity of these markets to external shocks in academic literature. These stock markets were previously analyzed in the context of portfolio diversification opportunity (see, e.g., [1, 2]). The very low degree of global integration of capital markets of Central and Eastern European countries in the pre-accession period was identified, among others, by Mateus [8], Maneschiold [9], and Nielsson [10]. An analysis of the sensitivity of the CEE-3 stock markets to global shocks during the dotcom crisis, which was conducted by Bein and Tuna [11], has indicated their calmness and excluded possibility of their significant reaction to negative news.

After announcement of the CEE-4 countries' EU membership, all stock markets in the region started to show similar level of volatility reactions to both negative and positive news that had the same magnitude. This finding has been interpreted as an increase in confidence for international investors after the announcement of the accession of countries of the Central and Eastern Europe to the European Union (see [12]). The accession of Poland, the Czech Republic, Hungary, and Slovakia to the EU on May 1, 2004, attracted the interest of many investors who had earlier refrained from buying assets of these countries due to corporate governance, political, and liquidity risks (see, e.g., [13–15]). An increase in the level of

*Stock Markets of the Visegrad Countries after Their Accession to the European Union DOI: http://dx.doi.org/10.5772/intechopen.92102*

integration resulted in a significant decrease in the benefits from using the portfolio diversification strategy (see [16]).

The impact of the subprime crisis on the performance of stock markets in CEE-4 countries has been broadly discussed in the economic literature. Very significant transmission of shocks to CEECs'stock markets during the subprime crisis was identified in numerous research studies (see, e.g., [17–19]). As Syllignakis and Kouretas [13] suggested, the contagion transmission from the major stock markets to capital markets of the CEECs in the period of financial turmoil as well as during the subprime crisis was due to increased financial liberalization and increased participation of foreign investors in these markets.

In turn, studies devoted to the role of the sovereign debt crisis in shaping stock market prices in the Central and Eastern Europe seem to be rare. The analysis of dynamic correlation coefficients conducted by Bein and Tuna [11] indicates that during the euro-area sovereign debt crisis, stock markets in the analyzed region have been highly correlated with the finance-led markets of GIIPS (Greece, Italy, Ireland, Portugal, and Spain) as well as with stock markets of the EU3 (France, Germany, and the United Kingdom). Moreover, significant spillover effect from capital markets of Italy, Ireland, Portugal, and Spain to stock markets of the Central and Eastern European countries has been identified. Differences in the level of reaction of stock markets in CEE-4 countries during the euro area sovereign debt crisis have been noticed as well. Due to the fact that Slovakia was a member of the euro area in 2010–2011, stock market in this country reacted stronger to positive and negative shocks. Moreover, the Polish stock market has shown a significantly higher level of conditional correlation than the Czech Republic and Hungary. Some results in the literature devoted to the sensitivity of capital markets in Visegrad countries to external shocks have pointed out their asymmetric reaction (see, e.g., [18, 20]). This finding has been interpreted as a problem of information asymmetry and the presence of agents with superior knowledge [21].

As Grabowski [20] and Moagar-Poladian et al. [22] have noticed, in the period of financial stability (2013–2019) in the case of Poland, the Czech Republic, and Hungary, the sensitivity of the stock market returns to external shocks became meaningfully weaker. In the case of the Slovakian stock market, drop in the level of integration was smaller (see, e.g., [23]). Moreover, after 2012, the evolution of the stock market indices in the CEE-4 countries has followed different paths. As a result of the lower level of market uncertainty, volatility spillovers have weakened. A within-group integration of stock markets of the CEE-4 countries has decreased significantly. Opportunities for portfolio diversification have increased with discrepancies between market returns observed after 2012 (see. [20]).

### **3. Methodology**

observed (see [5]). In particular, correlation between shocks in the Slovakian stock market and shocks in industrialized countries was very large. The outbreak of the euro area sovereign debt crisis resulted in a slight decrease in the level of correlation between stock markets in CEE-4 countries and mature capital markets. However, rates of return in the German stock market still strongly affected the rates of return in the Polish, Hungarian, Czech, and Slovakian stock market. After the introduction of the Outright Monetary Transactions program (OMT henceforth), the sensitivity of the stock market returns in Poland, the Czech Republic, and Hungary to external shocks decreased significantly. This phenomenon was not observed in the case of

Nevertheless, we still think that we can contribute to this body of literature. Previous studies devoted to the analysis of the performance of stock markets in Poland, the Czech Republic, Slovakia, and Hungary compared sensitivity of these markets to external and internal shocks in a priori defined subperiods. In most cases, the Winkler's [6] periodization of the global financial crisis was used. In this research study, we identify days of statistically significant breakpoints on the basis of the method proposed by Inclan and Tiao [7]. Stock markets of Poland, the Czech Republic, Slovakia, and Hungary are compared with regard to timing of significant breakpoints. Moreover, efficiency of stock markets in CEE-4 countries is compared across subperiods and across markets. Sensitivity of stock markets to external shocks is compared in the context of the volatility transmission and linkages between rates of return. Differences between resistance to shocks from the United States and Germany (main economy of the European Union) are analyzed as well. The obtained differences between Slovakia and noneuro-area member states should provide recommendations for policy makers of Poland, the Czech Republic, and Hungary in the context of the future accession of these countries to the euro area. This chapter has the following structure. In Section 2, the literature review is provided. In Section 3, the methodology is presented. In Section 4, findings from the empirical research are presented and discussed. The last section concludes the

Poor integration of capital markets in Poland, the Czech Republic, Hungary, and Slovakia in the twentieth century resulted in low level of interest in sensitivity of these markets to external shocks in academic literature. These stock markets were previously analyzed in the context of portfolio diversification opportunity (see, e.g., [1, 2]). The very low degree of global integration of capital markets of Central and Eastern European countries in the pre-accession period was identified, among others, by Mateus [8], Maneschiold [9], and Nielsson [10]. An analysis of the sensitivity of the CEE-3 stock markets to global shocks during the dotcom crisis, which was conducted by Bein and Tuna [11], has indicated their calmness and

After announcement of the CEE-4 countries' EU membership, all stock markets in the region started to show similar level of volatility reactions to both negative and positive news that had the same magnitude. This finding has been interpreted as an increase in confidence for international investors after the announcement of the accession of countries of the Central and Eastern Europe to the European Union (see [12]). The accession of Poland, the Czech Republic, Hungary, and Slovakia to the EU on May 1, 2004, attracted the interest of many investors who had earlier refrained from buying assets of these countries due to corporate governance, political, and liquidity risks (see, e.g., [13–15]). An increase in the level of

excluded possibility of their significant reaction to negative news.

Slovakia, which is a member of the euro area.

*Banking and Finance*

study.

**138**

**2. Literature review**

In order to identify different states of stock markets, the methodology proposed by Inclan and Tiao [7] and Inclan et al. [24] should be used. On the basis of the following statistics,

$$IT = D\_{k\*} \sqrt{T/2} \tag{1}$$

moments of significant changes in the unconditional variance were identified. In formula (1), the number of observations denoted by *T* and *Dk* <sup>∗</sup> is defined as follows:

$$D\_{k\*} = \begin{array}{c} \max \\ k \end{array} |D\_k|, \tag{2}$$

where

$$D\_k = \frac{C\_k}{C\_T} - \frac{k}{T},\tag{3}$$

where *n*<sup>1</sup> , *n*2, and *n*<sup>3</sup> denote the numbers of different types of series, and *n* is the

*Stock Markets of the Visegrad Countries after Their Accession to the European Union*

The statistic (2) is normally distributed with 0 mean and standard deviation 1. In order to test whether the series if securities follow random walk, the variance ratio tests (see [28]) are used. This test is based on the assumption that the variance of increments in a random walk is linearly independent. Variance ratio statistics are

In order to verify, whether the RW1 (assumption that increments of analyzed process are independent and identically distributed) null hypothesis is valid, the

*<sup>M</sup>*1ð Þ¼ *<sup>k</sup> VR k*ð Þ� <sup>1</sup> ffiffiffiffiffiffiffiffiffi

*<sup>ϕ</sup>*ð Þ¼ *<sup>k</sup>* 2 2ð Þ *<sup>k</sup>* � <sup>1</sup> ð Þ *<sup>k</sup>* � <sup>1</sup>

*<sup>M</sup>*2ð Þ¼ *<sup>k</sup> VR k*ð Þ� <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

*k*�1

2ð Þ *k* � *j k* � �<sup>2</sup>

2

*<sup>t</sup>*¼<sup>1</sup>ð Þ *rt* � *<sup>r</sup>*

Both statistics *M*1ð Þ*k* and *M*2ð Þ*k* follow standard normal distribution. If the null hypothesis is rejected, the analyzed time series is not random walk, and the capital

*rt*�*<sup>j</sup>* � *<sup>r</sup>* � �<sup>2</sup>

<sup>2</sup> h i<sup>2</sup> *:* (15)

**Π***ir<sup>t</sup>*�*<sup>i</sup>* þ *εt*, (16)

� � <sup>¼</sup> *<sup>H</sup>t*, (17)

*j*¼1

*<sup>t</sup>*¼*j*þ<sup>1</sup>ð Þ *rt* � *<sup>r</sup>*

P*<sup>T</sup>*

In order to analyze sensitivity of stock markets to external shocks, the following bivariate VAR(p)-AGDCC-GARCH(1,1) models (see [30]) will be

> *<sup>r</sup><sup>t</sup>* <sup>¼</sup> <sup>X</sup> *p*

> > *i*¼1

*E εtε<sup>T</sup> t*

dent but uncorrelated increments), the following statistic is used:

*<sup>ϕ</sup>*<sup>∗</sup> ð Þ¼ *<sup>k</sup>* <sup>X</sup>

P*<sup>T</sup>*

*δ <sup>j</sup>* ¼

market is not efficient (see, e.g., [29]).

In order to verify, whether the RW3 (it is assumed that the process has depen-

ð Þ *rt* þ *rt*�<sup>1</sup> þ … þ *rt*�*k*þ<sup>1</sup> *k* ∗ *S*<sup>2</sup> ð Þ *rt*

*:* (10)

*<sup>ϕ</sup>*ð Þ*<sup>k</sup>* <sup>p</sup> , (11)

<sup>3</sup>*kT :* (12)

*<sup>ϕ</sup>*<sup>∗</sup> ð Þ*<sup>k</sup>* <sup>p</sup> , (13)

*δ <sup>j</sup>* (14)

*VR k*ð Þ¼ *<sup>S</sup>*<sup>2</sup>

total number of series.

*DOI: http://dx.doi.org/10.5772/intechopen.92102*

calculated as follows:

following statistic is used:

where

where

and

considered:

**141**

*Ck* is the cumulative sum of squares.

Market efficiency belongs to the most important features describing financial markets in the area of information processing mechanisms. According to the concept introduced by Fama [25], on informationally efficient market prices always fully reflect the available information. As a result, it is not possible to achieve permanently superior returns on the basis of publicly available information, and the changes in financial asset prices are random. There are three main statistical methods of testing efficiency (autocorrelation test, runs test, and the test for the presence of unit root).

Testing the autocorrelation is based on the autocorrelation of the *k*th-order (ACF) coefficient, which is defined as follows:

$$\hat{\rho}\_k = \frac{\sum\_{t=k}^T (r\_t - \overline{r})(r\_{t-k} - \overline{r})}{\sum\_{t=1}^T (r\_t - \overline{r})^2},\tag{4}$$

where *rt* denotes a rate of return of a financial instrument. The lack of autocorrelation of the first order does not preclude the existence of autocorrelation of higher orders (see [26, 27]). Since the autocorrelation coefficients of higher order ignore the information provided by the observations between the first and last one in the period, analysis of capital market efficiency is conducted within the framework of the statistical analysis of the partial autocorrelation coefficients (PACFs).

The second method of testing efficiency is based on the run tests. The null hypothesis says that the changes of prices of securities are random. The test statistic is as follows:

$$U = \frac{K - E(\tilde{K})}{\mathcal{S}(\tilde{K})},\tag{5}$$

where *K* denotes count of empirical runs, *E K*~ � � denotes expected number of runs, while *S K*~ � � denotes standard deviation of the number of runs. If we consider two series (e.g., negative and non-negative), then the expectation and variance are calculated as follows:

$$E(\bar{K}) = \frac{2n\_1n\_2 + n}{n},\tag{6}$$

$$\mathcal{S}^2(\tilde{\mathcal{K}}) = \frac{2n\_1 n\_2 (2n\_1 n\_2 - n)}{(n - 1)n^2},\tag{7}$$

where *n*<sup>1</sup> and *n*<sup>2</sup> denote the numbers of different types of series, and *n* is the total number of series. If we consider three series (e.g., negative, zero, and positive), then the expectation and variance are calculated as follows:

$$E(\bar{K}) = n + 1 - \frac{\sum\_{j=1}^{3} n\_j^2}{n},\tag{8}$$

$$\mathcal{S}^2(\bar{\mathcal{K}}) = \frac{\sum\_{j=1}^3 n\_j^2 \left(\sum\_{j=1}^3 n\_j^2 + n + n^2\right) - 2n\sum\_{j=1}^3 n\_j^3 - n^3}{(n^2 - 1)n},\tag{9}$$

*Stock Markets of the Visegrad Countries after Their Accession to the European Union DOI: http://dx.doi.org/10.5772/intechopen.92102*

where *n*<sup>1</sup> , *n*2, and *n*<sup>3</sup> denote the numbers of different types of series, and *n* is the total number of series.

The statistic (2) is normally distributed with 0 mean and standard deviation 1.

In order to test whether the series if securities follow random walk, the variance ratio tests (see [28]) are used. This test is based on the assumption that the variance of increments in a random walk is linearly independent. Variance ratio statistics are calculated as follows:

$$VR(k) = \frac{\mathbf{S}^2(r\_t + r\_{t-1} + \dots + r\_{t-k+1})}{k \ast \mathbf{S}^2(r\_t)}.\tag{10}$$

In order to verify, whether the RW1 (assumption that increments of analyzed process are independent and identically distributed) null hypothesis is valid, the following statistic is used:

$$M\_1(k) = \frac{\text{VR}(k) - \mathbf{1}}{\sqrt{\phi(k)}},\tag{11}$$

where

where

*Banking and Finance*

is as follows:

runs, while *S K*~

calculated as follows:

*S*<sup>2</sup> *K*~ � � <sup>¼</sup>

**140**

*Ck* is the cumulative sum of squares.

(ACF) coefficient, which is defined as follows:

^*ρ<sup>k</sup>* ¼

where *K* denotes count of empirical runs, *E K*~

then the expectation and variance are calculated as follows:

*E K*~

P<sup>3</sup> *<sup>j</sup>*¼<sup>1</sup>*n*<sup>2</sup> *j* P<sup>3</sup> *<sup>j</sup>*¼<sup>1</sup>*n*<sup>2</sup>

� � <sup>¼</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> �

P*<sup>T</sup>*

P*<sup>T</sup>*

relation of the first order does not preclude the existence of autocorrelation of higher orders (see [26, 27]). Since the autocorrelation coefficients of higher order ignore the information provided by the observations between the first and last one in the period, analysis of capital market efficiency is conducted within the framework of the statistical analysis of the partial autocorrelation coefficients (PACFs). The second method of testing efficiency is based on the run tests. The null hypothesis says that the changes of prices of securities are random. The test statistic

*<sup>U</sup>* <sup>¼</sup> *<sup>K</sup>* � *<sup>E</sup> <sup>K</sup>*<sup>~</sup>

*S K*~

two series (e.g., negative and non-negative), then the expectation and variance are

*<sup>E</sup> <sup>K</sup>*~� � <sup>¼</sup> <sup>2</sup>*n*1*n*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*

*<sup>S</sup>*<sup>2</sup> *<sup>K</sup>*~� � <sup>¼</sup> <sup>2</sup>*n*1*n*2ð Þ <sup>2</sup>*n*1*n*<sup>2</sup> � *<sup>n</sup>*

where *n*<sup>1</sup> and *n*<sup>2</sup> denote the numbers of different types of series, and *n* is the total number of series. If we consider three series (e.g., negative, zero, and positive),

> *<sup>j</sup>* <sup>þ</sup> *<sup>n</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> � �

> > *n*ð Þ <sup>2</sup> � 1 *n*

P<sup>3</sup> *<sup>j</sup>*¼<sup>1</sup>*n*<sup>2</sup> *j*

> � 2*n* P<sup>3</sup> *<sup>j</sup>*¼<sup>1</sup>*n*<sup>3</sup> *<sup>j</sup>* � *<sup>n</sup>*<sup>3</sup>

� �

� � denotes standard deviation of the number of runs. If we consider

*Dk* <sup>¼</sup> *Ck CT* � *k*

Market efficiency belongs to the most important features describing financial markets in the area of information processing mechanisms. According to the concept introduced by Fama [25], on informationally efficient market prices always fully reflect the available information. As a result, it is not possible to achieve permanently superior returns on the basis of publicly available information, and the changes in financial asset prices are random. There are three main statistical methods of testing efficiency (autocorrelation test, runs test, and the test for the presence of unit root). Testing the autocorrelation is based on the autocorrelation of the *k*th-order

*<sup>t</sup>*¼*<sup>k</sup>*ð Þ *rt* � *<sup>r</sup>* ð Þ *rt*�*<sup>k</sup>* � *<sup>r</sup>*

*<sup>t</sup>*¼<sup>1</sup>ð Þ *rt* � *<sup>r</sup>*

where *rt* denotes a rate of return of a financial instrument. The lack of autocor-

*<sup>T</sup>* , (3)

<sup>2</sup> , (4)

� � , (5)

� � denotes expected number of

*<sup>n</sup>* , (6)

*<sup>n</sup>* , (8)

, (9)

ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>n</sup>*<sup>2</sup> , (7)

$$\phi(k) = \frac{2(2k-1)(k-1)}{3kT}.\tag{12}$$

In order to verify, whether the RW3 (it is assumed that the process has dependent but uncorrelated increments), the following statistic is used:

$$M\_2(k) = \frac{VR(k) - 1}{\sqrt{\phi^\*(k)}},\tag{13}$$

where

$$\phi^\*\left(k\right) = \sum\_{j=1}^{k-1} \left[\frac{2(k-j)}{k}\right]^2 \delta\_j \tag{14}$$

and

$$\delta\_{j} = \frac{\sum\_{t=j+1}^{T} (r\_t - \overline{r})^2 \left(r\_{t-j} - \overline{r}\right)^2}{\left[\sum\_{t=1}^{T} (r\_t - \overline{r})^2\right]^2}. \tag{15}$$

Both statistics *M*1ð Þ*k* and *M*2ð Þ*k* follow standard normal distribution. If the null hypothesis is rejected, the analyzed time series is not random walk, and the capital market is not efficient (see, e.g., [29]).

In order to analyze sensitivity of stock markets to external shocks, the following bivariate VAR(p)-AGDCC-GARCH(1,1) models (see [30]) will be considered:

$$r\_t = \sum\_{i=1}^p \Pi\_i r\_{t-i} + \mathbf{e}\_t,\tag{16}$$

$$E\left(\mathbf{e}\_t \mathbf{e}\_t^T\right) = \mathbf{H}\_t,\tag{17}$$

where *r<sup>t</sup>* is the following vector of rates of return on stock market indices:

$$r\_t = \begin{bmatrix} r\_t^{\text{VIS}} & r\_t^{\text{DEV}} \end{bmatrix}^T$$

*rVIS <sup>t</sup>* denotes the rate of return on stock market index in a Visegrad country (*VIS=PL,CZ,HU,SK*), and *r DEV <sup>t</sup>* denotes the rate of return on a mature stock market index (*DEV = DE,US*). The covariance matrix is decomposed as follows:

$$H\_t = \mathbf{D}\_t \mathbf{R}\_t \mathbf{D}\_t,\tag{18}$$

In the first step, the methodology proposed by Inclan and Tiao [7] and Inclan et al. [24] is used in order to identify dates of significant changes in the unconditional variance for Poland, the Czech Republic, Hungary, and Slovakia (**Table 1**). Results of the analysis indicate that significant changes in volatilities of the rates of return were observed in similar days in all markets. Significant breakpoint in variance during the financial turmoil and short before the Lehman Brothers bankruptcy is in line with expectations. The second breakpoint date is observed during the euro area sovereign debt crisis. However, in the case of Slovakia, a significant change was observed short before the announcement of the OMT program. Difference between Slovakia and noneuro-area members does not come as a surprise, since the performance of financial markets in Slovakia was strongly linked to the situation in the euro area. The announcement of the OMT program turned out to have the long-term impact on markets in the euro area, so the obtained result confirms findings of other studies (see [31, 32]). Moreover, short after the Brexit referendum, significant breakpoint in variance is observed. This result confirms findings obtained by Kurecic and Kokotovic [33], who have also noticed a significant increase in volatility on stock markets after this event. Results of the British referendum could provide information about a threat of an illiberal turn all over the world. As a result, level of trust in financial markets decreased significantly, which was reflected by the appearance of the home bias phenomenon. All in all, in the case of all four countries, four in terms of volatility four states can be distinguished.

*Stock Markets of the Visegrad Countries after Their Accession to the European Union*

In the next step, hypothesis concerning efficiency of stock markets is verified on

Results from **Tables 2**–**5** indicate that stock markets in four analyzed countries were efficient in most of analyzed subperiods. However, in some cases, conclusions depend on the used methodology. The lowest level of efficiency was observed during the subprime crisis in the United States. This result is not surprising, since after the Lehman Brothers bankruptcy, very large panic on stock markets was observed. As a result of this panic, efficiency of financial markets decreased significantly. Some studies set hypothesis that the phenomenon of I(2)-ness of prices of financial instruments in crisis periods exists (see, e.g., [34]). Inefficiency of stock markets in the crisis period may confirm validity of this hypothesis. It turns out that stock markets of countries of the Central and Eastern Europe differ with regard to efficiency. For example, after the Lehman Brothers bankruptcy, stock markets in Hungary and Poland turned out to be inefficient, while the H0 hypothesis about efficiency was not rejected in the case of Slovakia and the Czech Republic. Differences across subperiods suggest that it is difficult to infer about efficiency in the

the basis of three methods for all four subperiods and four countries. Results of verification for Poland are presented in **Table 2**. Results of verification for the Czech Republic are presented in **Table 3**. Results of verification for Hungary and

**Republic**

**Hungary Slovakia Event**

crisis

sovereign debt crisis

referendum

2008-09-12 2008-09-04 2008-09-12 2008-09-12 The US subprime

2011-12-21 2011-12-23 2012-01-23 2012-07-23 The euro area

2016-06-27 2016-07-18 2016-06-29 2016-06-28 The Brexit

Slovakia are presented in **Tables 4** and **5**.

*DOI: http://dx.doi.org/10.5772/intechopen.92102*

**Country Poland Czech**

*Results of the analysis of breakpoints in variance.*

Dates of breakpoint in variances

**Table 1.**

**143**

where the matrix *D<sup>t</sup>* consists of squared roots of variances of shocks:

$$\mathbf{D}\_t = \text{diag}\left[\sqrt{h\_t^{\text{VIS,VIS}}} \quad \sqrt{h\_t^{\text{DEV,DEV}}}\right]. \tag{19}$$

These variances of shocks are modeled according to the following GARCH(1,1) model:

$$h\_t^{n,n} = a\_{0,n} + a\_{1,n} \left(\varepsilon\_{t-1}^n\right)^2 + \beta\_{1,n} h\_{t-1}^{n,n}, n = \text{VIS}, \text{DEV}.\tag{20}$$

Correlations between shocks change in time and depend on negative and positive shocks in the following way:

$$\mathbf{R}\_{t} = \left(\text{diag}(\mathbf{Q}\_{t})\right)^{-1/2} \mathbf{Q}\_{t} \left(\text{diag}(\mathbf{Q}\_{t})\right)^{-1/2},\tag{21}$$

where

$$\mathbf{Q}\_{t} = \left(\mathbf{1} - \boldsymbol{\tilde{a}}\_{1} - \boldsymbol{\tilde{\beta}}\_{1}\right)\overline{\mathbf{Q}} + \boldsymbol{\tilde{\gamma}}\_{1}\left(\overline{\mathbf{Q}} - \overline{\mathbf{Q}}^{-}\right) + \boldsymbol{\tilde{a}}\_{1}\mathbf{u}\_{t-1}\boldsymbol{\mathfrak{u}}\_{t-1}^{T} + \boldsymbol{\tilde{\beta}}\_{1}\mathbf{Q}\_{t} + \boldsymbol{\tilde{\gamma}}\_{1}\boldsymbol{\mathfrak{u}}\_{t-1}^{-}\left(\boldsymbol{\mathfrak{u}}\_{t-1}^{-}\right)^{T}.\tag{22}$$

The elements of vector *u<sup>t</sup>* are defined as follows:

$$u\_t^n = \frac{\varepsilon\_t^n}{\sqrt{h\_t^{n,n}}},\tag{23}$$

where *n = VIS,DEV* and *u*� *<sup>t</sup>*�<sup>1</sup> consists of zero-threshold standardized errors, while the matrices *Q* and *Q*� denote the unconditional covariance matrices of vectors *u<sup>t</sup>*�<sup>1</sup> and *u*� *<sup>t</sup>*�1, respectively.

As a robustness check, parameters of the asymmetric VAR(p)-GARCH-BEKK model (see [30]) will be estimated. In this model, the covariance matrix evolves according to the following formula:

$$\mathbf{H}\_{t} = \mathbf{C}\mathbf{C}^{T} + \mathbf{A}\mathbf{e}\_{t-1}\mathbf{e}\_{t-1}^{T}\mathbf{A}^{T} + \mathbf{B}\mathbf{H}\_{t-1}\mathbf{B}^{T} + \mathbf{D}\mathbf{f}\_{t-1}\mathbf{f}\_{t-1}^{T}\mathbf{D}^{T},\tag{24}$$

#### **4. Results and discussion**

In the empirical research, we use daily data covering period from May 2004 to December 2019. Logarithmic rates of return on WIG (acronym of the Warsaw Stock Index in Polish), BUX (acronym of the Budapest Stock Index in Hungarian), PX (acronym of the Prague Stock Exchange in Czech), SAX (acronym of the Slovakian Stock Index in Slovakian), DAX (acronym of the German Stock Index in German language), and S&P500 (Standard and Poor's 500) are used.

#### *Stock Markets of the Visegrad Countries after Their Accession to the European Union DOI: http://dx.doi.org/10.5772/intechopen.92102*

In the first step, the methodology proposed by Inclan and Tiao [7] and Inclan et al. [24] is used in order to identify dates of significant changes in the unconditional variance for Poland, the Czech Republic, Hungary, and Slovakia (**Table 1**).

Results of the analysis indicate that significant changes in volatilities of the rates of return were observed in similar days in all markets. Significant breakpoint in variance during the financial turmoil and short before the Lehman Brothers bankruptcy is in line with expectations. The second breakpoint date is observed during the euro area sovereign debt crisis. However, in the case of Slovakia, a significant change was observed short before the announcement of the OMT program. Difference between Slovakia and noneuro-area members does not come as a surprise, since the performance of financial markets in Slovakia was strongly linked to the situation in the euro area. The announcement of the OMT program turned out to have the long-term impact on markets in the euro area, so the obtained result confirms findings of other studies (see [31, 32]). Moreover, short after the Brexit referendum, significant breakpoint in variance is observed. This result confirms findings obtained by Kurecic and Kokotovic [33], who have also noticed a significant increase in volatility on stock markets after this event. Results of the British referendum could provide information about a threat of an illiberal turn all over the world. As a result, level of trust in financial markets decreased significantly, which was reflected by the appearance of the home bias phenomenon. All in all, in the case of all four countries, four in terms of volatility four states can be distinguished.

In the next step, hypothesis concerning efficiency of stock markets is verified on the basis of three methods for all four subperiods and four countries. Results of verification for Poland are presented in **Table 2**. Results of verification for the Czech Republic are presented in **Table 3**. Results of verification for Hungary and Slovakia are presented in **Tables 4** and **5**.

Results from **Tables 2**–**5** indicate that stock markets in four analyzed countries were efficient in most of analyzed subperiods. However, in some cases, conclusions depend on the used methodology. The lowest level of efficiency was observed during the subprime crisis in the United States. This result is not surprising, since after the Lehman Brothers bankruptcy, very large panic on stock markets was observed. As a result of this panic, efficiency of financial markets decreased significantly. Some studies set hypothesis that the phenomenon of I(2)-ness of prices of financial instruments in crisis periods exists (see, e.g., [34]). Inefficiency of stock markets in the crisis period may confirm validity of this hypothesis. It turns out that stock markets of countries of the Central and Eastern Europe differ with regard to efficiency. For example, after the Lehman Brothers bankruptcy, stock markets in Hungary and Poland turned out to be inefficient, while the H0 hypothesis about efficiency was not rejected in the case of Slovakia and the Czech Republic. Differences across subperiods suggest that it is difficult to infer about efficiency in the


#### **Table 1.** *Results of the analysis of breakpoints in variance.*

where *r<sup>t</sup>* is the following vector of rates of return on stock market indices:

*VIS <sup>t</sup> r DEV t* � �*<sup>T</sup>*

*<sup>t</sup>* denotes the rate of return on stock market index in a Visegrad country

*<sup>t</sup>* denotes the rate of return on a mature stock market

*hDEV*,*DEV t*

*<sup>Q</sup><sup>t</sup> diag <sup>Q</sup><sup>t</sup>* ð Þ ð Þ �1*=*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*~1*Q<sup>t</sup>* <sup>þ</sup> <sup>~</sup>*γ*1*u*�

*<sup>t</sup>*�<sup>1</sup> consists of zero-threshold standardized errors,

<sup>p</sup> , (23)

*H<sup>t</sup>* ¼ *DtRtDt*, (18)

*:* (19)

, (21)

*<sup>t</sup>*�<sup>1</sup>*D<sup>T</sup>*, (24)

*:* (22)

*<sup>t</sup>*�<sup>1</sup> *u*� *t*�1 � �*<sup>T</sup>*

*<sup>t</sup>*�1, *n* ¼ *VIS*, *DEV:* (20)

*r<sup>t</sup>* ¼ *r*

index (*DEV = DE,US*). The covariance matrix is decomposed as follows:

where the matrix *D<sup>t</sup>* consists of squared roots of variances of shocks:

*t*�1

*un <sup>t</sup>* <sup>¼</sup> *<sup>ε</sup><sup>n</sup> t* ffiffiffiffiffiffiffiffi *h<sup>n</sup>*,*<sup>n</sup> t*

while the matrices *Q* and *Q*� denote the unconditional covariance matrices of

As a robustness check, parameters of the asymmetric VAR(p)-GARCH-BEKK model (see [30]) will be estimated. In this model, the covariance matrix evolves

In the empirical research, we use daily data covering period from May 2004 to December 2019. Logarithmic rates of return on WIG (acronym of the Warsaw Stock Index in Polish), BUX (acronym of the Budapest Stock Index in Hungarian), PX (acronym of the Prague Stock Exchange in Czech), SAX (acronym of the Slovakian Stock Index in Slovakian), DAX (acronym of the German Stock Index in

German language), and S&P500 (Standard and Poor's 500) are used.

*<sup>t</sup>*�<sup>1</sup>*A<sup>T</sup>* <sup>þ</sup> *BH<sup>t</sup>*�<sup>1</sup>*B<sup>T</sup>* <sup>þ</sup> *<sup>D</sup>ξ<sup>t</sup>*�<sup>1</sup>*ξ<sup>T</sup>*

*<sup>R</sup><sup>t</sup>* <sup>¼</sup> *diag <sup>Q</sup><sup>t</sup>* ð Þ ð Þ �1*=*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *hVIS*,*VIS t*

These variances of shocks are modeled according to the following GARCH(1,1)

� �<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*1,*nh<sup>n</sup>*,*<sup>n</sup>*

Correlations between shocks change in time and depend on negative and posi-

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h i q

*D<sup>t</sup>* ¼ *diag*

*<sup>t</sup>* <sup>¼</sup> *<sup>α</sup>*0,*<sup>n</sup>* <sup>þ</sup> *<sup>α</sup>*1,*<sup>n</sup> <sup>ε</sup><sup>n</sup>*

� �*<sup>Q</sup>* <sup>þ</sup> <sup>~</sup>*γ*<sup>1</sup> *<sup>Q</sup>* � *<sup>Q</sup>* � � � <sup>þ</sup> *<sup>α</sup>*~1*u<sup>t</sup>*�<sup>1</sup>*u<sup>T</sup>*

The elements of vector *u<sup>t</sup>* are defined as follows:

*<sup>t</sup>*�1, respectively.

*<sup>H</sup><sup>t</sup>* <sup>¼</sup> *CC<sup>T</sup>* <sup>þ</sup> *<sup>A</sup>ε<sup>t</sup>*�<sup>1</sup>*ε<sup>T</sup>*

*rVIS*

*Banking and Finance*

model:

where

*<sup>Q</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>α</sup>*~<sup>1</sup> � *<sup>β</sup>*~<sup>1</sup>

vectors *u<sup>t</sup>*�<sup>1</sup> and *u*�

**142**

(*VIS=PL,CZ,HU,SK*), and *r DEV*

*hn*,*<sup>n</sup>*

tive shocks in the following way:

where *n = VIS,DEV* and *u*�

according to the following formula:

**4. Results and discussion**


#### **Table 2.**

*Results of testing efficiency of the Polish stock market.*


• before the Lehman Brothers bankruptcy,

**Description of subperiod Dates Variance**

*Stock Markets of the Visegrad Countries after Their Accession to the European Union*

Before the subprime crisis 2004-05-03 to

• after the announcement of the OMT program.

OMT program, and

Subprime crisis and the euro area

The last period after the Brexit

Calming down after crisis and before the

*DOI: http://dx.doi.org/10.5772/intechopen.92102*

sovereign debt crisis

Brexit referendum

referendum

**Table 5.**

average values of correlations

First subperiod

Second subperiod

Third subperiod

**Table 6.**

**145**

• between the Lehman Brothers bankruptcy and before the announcement of the

2008-09-12

2008-09-15 to 2012-01-23

2012-01-24 to 2016-06-29

2016-06-30 to 2019-12-30

*Results of testing efficiency of the Slovakian stock market.* P *values of statistical tests are provided.*

**ratio test**

**Autocorrelation function**

0.3055 0.079 0.24

0.0211 0.003 0.13

0.8756 0.005 0.88

0.4549 0.274 0.20

**Run test**

**Table 6** presents estimates of parameters reflecting impact of rates of return for developed markets on the rates of return for Visegrad stock markets, while the table

**WIG BUX PX SAX**

Results of the analysis indicate the strength of the impact of developed stock markets on Visegrad stock markets differed across subperiods (**Table 7**). The strongest linkages were observed in the crisis period. Between 2008 and 2012, shocks generated by stock markets in Visegrad countries were strongly correlated with shocks generated by stock markets in Germany and the United States. A significant drop in comovements of stock markets in Poland, the Czech Republic, Hungary, and capital markets of Germany and the United States was observed after the announcement of the OMT program. In the stable period, investors got knowledge which classes of assets were riskier or safer. Moreover, the illiberal turn in the Central and Eastern Europe could have impact on sensitivity of stock markets in Poland, the Czech Republic, and Hungary on external shocks (see, e.g., [35]) In the case of the correlation between shocks associated with SAX and DAX, a decrease in correlation was not so large, which should be attributed with participation of

DAX 0.549 0.705 0.237 0.187 S&P500 0.313 0.743 0.226 0.278

DAX 0.643 0.792 0.289 0.367 S&P500 0.421 0.851 0.113 0.389

DAX 0.439 0.266 0.125 0.191 S&P500 0.206 0.299 0.098 0.235

*Impact of rates of return of developed markets on rates of return of Visegrad countries.*

#### **Table 3.**

*Results of testing efficiency of the Czech stock market.*


#### **Table 4.**

*Results of testing efficiency of the Hungarian stock market.* P *values of statistical tests are provided.*

whole period, and some financial markets may be efficient in one period and inefficient in the other one.

In the last step, financial markets of the four analyzed countries are compared with regard to sensitivity to shocks from Germany and the United States. Multivariate GARCH models take into account time-varying volatilities. Therefore, division into subperiods according to **Table 1** is not necessary. In turn, analysis will be conducted in three large subperiods:


## **Table 5.**

*Results of testing efficiency of the Slovakian stock market.* P *values of statistical tests are provided.*


**Table 6** presents estimates of parameters reflecting impact of rates of return for developed markets on the rates of return for Visegrad stock markets, while the table average values of correlations

Results of the analysis indicate the strength of the impact of developed stock markets on Visegrad stock markets differed across subperiods (**Table 7**). The strongest linkages were observed in the crisis period. Between 2008 and 2012, shocks generated by stock markets in Visegrad countries were strongly correlated with shocks generated by stock markets in Germany and the United States. A significant drop in comovements of stock markets in Poland, the Czech Republic, Hungary, and capital markets of Germany and the United States was observed after the announcement of the OMT program. In the stable period, investors got knowledge which classes of assets were riskier or safer. Moreover, the illiberal turn in the Central and Eastern Europe could have impact on sensitivity of stock markets in Poland, the Czech Republic, and Hungary on external shocks (see, e.g., [35]) In the case of the correlation between shocks associated with SAX and DAX, a decrease in correlation was not so large, which should be attributed with participation of


#### **Table 6.**

*Impact of rates of return of developed markets on rates of return of Visegrad countries.*

whole period, and some financial markets may be efficient in one period and

*Results of testing efficiency of the Hungarian stock market.* P *values of statistical tests are provided.*

**Description of subperiod Dates Variance**

**Description of subperiod Dates Variance**

**Description of subperiod Dates Variance**

Before the subprime crisis 2004-05-03 to

Before the subprime crisis 2004-05-03 to

2008-09-12

2008-09-15 to 2011-12-21

2011-12-22 to 2016-07-27

2016-07-28 to 2019-12-30

2008-09-04

2008-09-05 to 2011-12-23

2011-12-24 to 2016-07-18

2016-07-19 to 2019-12-30

2008-09-12

2008-09-15 to 2012-01-23

2012-01-24 to 2016-06-29

2016-06-30 to 2019-12-30

Before the subprime crisis 2004-05-03 to

Subprime crisis and the euro area

The last period after the Brexit

P *values of statistical tests are provided.*

Subprime crisis and the euro area

The last period after the Brexit

P *values of statistical tests are provided.*

Subprime crisis and the euro area

The last period after the Brexit

Calming down after crisis and before the

sovereign debt crisis

Brexit referendum

referendum

**Table 4.**

**144**

Calming down after crisis and before the

*Results of testing efficiency of the Czech stock market.*

sovereign debt crisis

Brexit referendum

referendum

**Table 3.**

Calming down after crisis and before the

*Results of testing efficiency of the Polish stock market.*

sovereign debt crisis

*Banking and Finance*

Brexit referendum

referendum

**Table 2.**

**ratio test**

**ratio test**

**ratio test**

**Autocorrelation function**

0.9833 0.362 0.44

0.0488 0.012 0.64

0.9715 0.306 0.76

0.7031 0.248 0.96

**Autocorrelation function**

0.9050 0.403 0.64

0.9785 0.717 0.31

0.6140 0.071 0.64

0.7835 0.367 0.45

**Autocorrelation function**

0.3055 0.079 0.24

0.0211 0.003 0.13

0.8756 0.005 0.88

0.4549 0.274 0.20

**Run test**

**Run test**

**Run test**

In the last step, financial markets of the four analyzed countries are compared with regard to sensitivity to shocks from Germany and the United States. Multivariate GARCH models take into account time-varying volatilities. Therefore, division into subperiods according to **Table 1** is not necessary. In turn, analysis will be

inefficient in the other one.

conducted in three large subperiods:


#### **Table 7.**

*Correlations between shocks generated.*

Slovakia in the euro area. However, rates of return on WIG and BUX turned out to be more sensitive to changes in rates of return on DAX in particular in years 2004– 2012. This may be due to large interactions between Germany and two economies of the Central and Eastern Europe (Polish economy and Hungarian economy).
